Algebraic conformal quantum field theory in perspective

TL;DR

This work surveys conformal quantum field theory through the lens of axiomatic and algebraic quantum field theory, with a focus on two dimensions where rigorous constructions and classifications are tractable. It maps the landscape from Wightman and Haag–Kastler formalisms to chiral CFT nets on the circle, detailing how superselection sectors, modular tensor categories, and the Virasoro/affine algebras organize representations and model-building. The survey covers concrete chiral constructions (free fields, bosonization/fermionization, orbifolds, simple currents, cosets) and nonconformal algebraic methods (Q-systems, Borchers triples, holography, phase boundaries), illustrating how the algebraic framework unifies diverse approaches and sheds light on 4D QFT. It also discusses the special features of 2D CFT, such as braiding and modularity, and what these imply for potential transfer of insights to higher dimensions.

Abstract

Conformal quantum field theory is reviewed in the perspective of Axiomatic, notably Algebraic QFT. This theory is particularly developped in two spacetime dimensions, where many rigorous constructions are possible, as well as some complete classifications. The structural insights, analytical methods and constructive tools are expected to be useful also for four-dimensional QFT.